Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2001-02-13
Can.J.Phys. 67 (1989) 919
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
16 pages
Scientific paper
The historical postulates for the point mass are shown to be satisfied by an infinity of space-times, differing as to the limiting acceleration of a radially approaching test particle. Taking this limit to be infinite gives Schwarzschild's result, which for a point mass at x = y = z = 0 has C(0+) = a^2, where a = 2m and C(r) denotes the coefficient of the angular terms in the polar form of the metric. Hilbert's derivation used the variable r* =[C(0+)]^1/2. For Hilbert, however, C was unknown, and thus could not be used to determine r*(0). Instead he asserted, in effect, that r* = (x^2 + y^2 + z^2)^1/2, which places the point mass at r* = 0. Unfortunately, this differs from the value (a) obtained by substituting Schwarzschild's C into the expression for r*(0), and since C(0+) is a scaler invariant, it follows that Hilbert's assertion is invalid. Owing to this error, in each spatial section of Hilbert's space-time, the boundary (r* = a) corresponding to r = 0 is no longer a point, but a two-sphere. This renders his space-time analytically extendible, and as shown by Kruskal and Fronsdal, its maximal extension contains a black hole. Thus the Kruskal-Fronsdal black hole is merely an artifact of Hilbert's error.
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